Hans-Gill, R. J. ; Raka, Madhu ; Sehmi, Ranjeet
(2010)
*Estimates on conjectures of Minkowski and woods*
Indian Journal of Pure and Applied Mathematics, 41
(4).
pp. 595-606.
ISSN 0019-5588

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Official URL: http://www.springerlink.com/content/2g755m73t1tn82...

Related URL: http://dx.doi.org/10.1007/s13226-010-0034-9

## Abstract

Let R^{n} be the n-dimensional Euclidean space. Let Λ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of Λ other than the origin O and has n linearly independent points of Λ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in R^{n} of radius √n/4 contains a point of Λ. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il'in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.

Item Type: | Article |
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Source: | Copyright of this article belongs to Indian National Science Academy. |

Keywords: | Lattice; Covering; Non-homogeneous; Product of linear forms; Critical determinant; Korkine and Zolotareff reduction; Hermite's constant; Centre density |

ID Code: | 12285 |

Deposited On: | 10 Nov 2010 05:16 |

Last Modified: | 03 Jun 2011 06:26 |

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